This is my understanding: say you have a spring which is just two positive particles. When you compress it, bringing the particles closer, you're storing energy in the EM field (E²+B²) near the particles. When you accelerate the spring, you're also accelerating the energy, which requires extra energy. This feels the same as the spring becoming heavier.

This is just a microscopic explanation for the general principle that energy has mass. I've never done the math to confirm that the extra energy to accelerate the compressed spring corresponds to the extra mass from stored energy.